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| Mirrors > Home > MPE Home > Th. List > Mathboxes > m1expevenALTV | Structured version Visualization version GIF version | ||
| Description: Exponentiation of -1 by an even power. (Contributed by Glauco Siliprandi, 29-Jun-2017.) (Revised by AV, 6-Jul-2020.) |
| Ref | Expression |
|---|---|
| m1expevenALTV | ⊢ (𝑁 ∈ Even → (-1↑𝑁) = 1) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqeq1 2741 | . . . 4 ⊢ (𝑛 = 𝑁 → (𝑛 = (2 · 𝑖) ↔ 𝑁 = (2 · 𝑖))) | |
| 2 | 1 | rexbidv 3179 | . . 3 ⊢ (𝑛 = 𝑁 → (∃𝑖 ∈ ℤ 𝑛 = (2 · 𝑖) ↔ ∃𝑖 ∈ ℤ 𝑁 = (2 · 𝑖))) |
| 3 | dfeven4 47625 | . . 3 ⊢ Even = {𝑛 ∈ ℤ ∣ ∃𝑖 ∈ ℤ 𝑛 = (2 · 𝑖)} | |
| 4 | 2, 3 | elrab2 3695 | . 2 ⊢ (𝑁 ∈ Even ↔ (𝑁 ∈ ℤ ∧ ∃𝑖 ∈ ℤ 𝑁 = (2 · 𝑖))) |
| 5 | oveq2 7439 | . . . . 5 ⊢ (𝑁 = (2 · 𝑖) → (-1↑𝑁) = (-1↑(2 · 𝑖))) | |
| 6 | neg1cn 12380 | . . . . . . . . 9 ⊢ -1 ∈ ℂ | |
| 7 | 6 | a1i 11 | . . . . . . . 8 ⊢ (𝑖 ∈ ℤ → -1 ∈ ℂ) |
| 8 | neg1ne0 12382 | . . . . . . . . 9 ⊢ -1 ≠ 0 | |
| 9 | 8 | a1i 11 | . . . . . . . 8 ⊢ (𝑖 ∈ ℤ → -1 ≠ 0) |
| 10 | 2z 12649 | . . . . . . . . 9 ⊢ 2 ∈ ℤ | |
| 11 | 10 | a1i 11 | . . . . . . . 8 ⊢ (𝑖 ∈ ℤ → 2 ∈ ℤ) |
| 12 | id 22 | . . . . . . . 8 ⊢ (𝑖 ∈ ℤ → 𝑖 ∈ ℤ) | |
| 13 | expmulz 14149 | . . . . . . . 8 ⊢ (((-1 ∈ ℂ ∧ -1 ≠ 0) ∧ (2 ∈ ℤ ∧ 𝑖 ∈ ℤ)) → (-1↑(2 · 𝑖)) = ((-1↑2)↑𝑖)) | |
| 14 | 7, 9, 11, 12, 13 | syl22anc 839 | . . . . . . 7 ⊢ (𝑖 ∈ ℤ → (-1↑(2 · 𝑖)) = ((-1↑2)↑𝑖)) |
| 15 | neg1sqe1 14235 | . . . . . . . . 9 ⊢ (-1↑2) = 1 | |
| 16 | 15 | oveq1i 7441 | . . . . . . . 8 ⊢ ((-1↑2)↑𝑖) = (1↑𝑖) |
| 17 | 1exp 14132 | . . . . . . . 8 ⊢ (𝑖 ∈ ℤ → (1↑𝑖) = 1) | |
| 18 | 16, 17 | eqtrid 2789 | . . . . . . 7 ⊢ (𝑖 ∈ ℤ → ((-1↑2)↑𝑖) = 1) |
| 19 | 14, 18 | eqtrd 2777 | . . . . . 6 ⊢ (𝑖 ∈ ℤ → (-1↑(2 · 𝑖)) = 1) |
| 20 | 19 | adantl 481 | . . . . 5 ⊢ ((𝑁 ∈ ℤ ∧ 𝑖 ∈ ℤ) → (-1↑(2 · 𝑖)) = 1) |
| 21 | 5, 20 | sylan9eqr 2799 | . . . 4 ⊢ (((𝑁 ∈ ℤ ∧ 𝑖 ∈ ℤ) ∧ 𝑁 = (2 · 𝑖)) → (-1↑𝑁) = 1) |
| 22 | 21 | rexlimdva2 3157 | . . 3 ⊢ (𝑁 ∈ ℤ → (∃𝑖 ∈ ℤ 𝑁 = (2 · 𝑖) → (-1↑𝑁) = 1)) |
| 23 | 22 | imp 406 | . 2 ⊢ ((𝑁 ∈ ℤ ∧ ∃𝑖 ∈ ℤ 𝑁 = (2 · 𝑖)) → (-1↑𝑁) = 1) |
| 24 | 4, 23 | sylbi 217 | 1 ⊢ (𝑁 ∈ Even → (-1↑𝑁) = 1) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∃wrex 3070 (class class class)co 7431 ℂcc 11153 0cc0 11155 1c1 11156 · cmul 11160 -cneg 11493 2c2 12321 ℤcz 12613 ↑cexp 14102 Even ceven 47611 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-n0 12527 df-z 12614 df-uz 12879 df-seq 14043 df-exp 14103 df-even 47613 |
| This theorem is referenced by: m1expoddALTV 47635 |
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